WASHINGTON—A panel of Senate lawmakers on Tuesday voted along party lines to advance the nomination of Mark Calabria, President Trump’s pick to help overhaul the way many Americans finance their home purchases, setting up his likely confirmation later this spring.

The Senate Banking Committee voted 13-12 to approve Mr. Calabria’s nomination to become director of the Federal Housing Finance Agency, which regulates the mortgage-finance giants Fannie Mae and Freddie Mac.